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Title:
Constructing Dembowski–Ostrom permutation polynomials from upper triangular matrices
Authors: Yuyin Yu, Yanbin Zheng, Yongqiang Li, Jingang Liu
Abstract:We establish a one-to-one correspondence between Dembowski-Ostrom (DO) polynomials and upper triangular matrices. Based on this correspondence, we give a bijection between DO permutation polynomials and a special class of upper triangular matrices, and construct a new batch of DO permutation polynomials. To the best of our knowledge, almost all other known DO permutation polynomials are located in finite fields of \mathbb{F}_{2^n}, where n contains odd factors (see Table 1). However, there are no restrictions on n in our results, and especially the case of n=2^m has not been studied in the literature. For example, we provide a simple necessary and sufficient condition to determine when \gamma\, Tr(\theta_{i}x)Tr(\theta_{j}x) + x is a DO permutation polynomial. In addition, when the upper triangular matrix degenerates into a diagonal matrix and the elements on the main diagonal form a basis of \mathbb{F}_{q^{n}} over \mathbb{F}_{q}, this diagonal matrix corresponds to all linearized permutation polynomials. In a word, we construct several new DO permutation polynomials, and our results can be viewed as an extension of linearized permutation polynomials.
ePrint: https://eprint.iacr.org/2024/1824
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